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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1248, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,3,6,5]))
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pari:[g,chi] = znchar(Mod(1241,1248))
\(\chi_{1248}(137,\cdot)\)
\(\chi_{1248}(665,\cdot)\)
\(\chi_{1248}(713,\cdot)\)
\(\chi_{1248}(1241,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((703,1093,833,769)\) → \((1,i,-1,e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1248 }(1241, a) \) |
\(1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
